9.2 Up To Diffeomorphism

The mind
of man is more intuitive than logical, and comprehends more than it can
coordinate.

Vauvenargues,
1746

Einstein seems to have been
strongly wedded to the concept of the continuum described by partial
differential equations as the only satisfactory framework for physics. He was
certainly not the first to hold this view. For example, in 1860 Riemann wrote

As is well known, physics
became a science only after the invention of differential calculus. It was
only after realizing that natural phenomena are continuous that attempts to
construct abstract models were successful… In the first period, only certain
abstract cases were treated: the mass of a body was considered to be
concentrated at its center, the planets were mathematical points… so the
passage from the infinitely near to the finite was made only in one variable,
the time [i.e., by means of total differential equations]. In general,
however, this passage has to be done in several variables… Such passages lead
to partial differential equations… In all physical theories, partial
differential equations constitute the only verifiable basis. These facts,
established by induction, must also hold a priori. True basic laws can
only hold in the small and must be formulated as partial differential
equations.

Compare this with Einstein’s
comments (see Section 3.2) over 70 years later about the unsatisfactory
dualism inherent in Lorentz’s theory, which expressed the laws of motion of
particles in the form of total differential equations while describing the
electromagnetic field by means of partial differential equations. Interestingly,
Riemann asserted that the continuous nature of physical phenomena was
“established by induction”, but immediately went on to say it must also hold a
priori, referring somewhat obscurely to the idea that “true basic laws
can only hold in the infinitely small”. He may have been trying to convey by
these words his rejection of “action at a distance”. Einstein attributed this
insight to the special theory of relativity, but of course the Newtonian concept
of instantaneous action at a distance had always been viewed skeptically, so
it isn’t surprising that Riemann in 1860 – like his contemporary Maxwell –
adopted the impossibility of distant action as a fundamental principle. (It’s
interesting the consider whether Einstein might have taken this, rather than
the invariance of light speed, as one of the founding principles of special
relativity, since it immediately leads to the impossibility of rigid bodies,
etc.) In his autobiographical notes (1949) Einstein wrote

There is no such thing as
simultaneity of distant events; consequently, there is also no such thing as
immediate action at a distance in the sense of Newtonian mechanics. Although
the introduction of actions at a distance, which propagate at the speed of
light, remains feasible according to this theory, it appears unnatural; for
in such a theory there could be no reasonable expression for the principle of
conservation of energy. It therefore appears unavoidable that physical
reality must be described in terms of continuous functions in space.

It’s worth noting that
while Riemann and Maxwell had expressed their objections in terms of “action
at a (spatial) distance”, Einstein can justly claim that special relativity
revealed that the actual concept to be rejected was instantaneous
action at a distance. He acknowledge that “distant action” propagating at the
speed of light – which is to say, action over null intervals – is remains
feasible. In fact, one could argue that such “distant action” was made more
feasible by special relativity, especially in the context of Minkowski’s
spacetime, in which the null (light-like) intervals have zero absolute
magnitude. For any two light-like separated events there exist perfectly
valid systems of inertial coordinates in terms of which both the
spatial and the temporal measures of distance are arbitrarily small. It
doesn’t seem to have troubled Einstein (nor many later scientists) that the
existence of non-trivial null intervals potentially undermines the
identification of the topology of pseudo-metrical spacetime with that of a
true metric space. Thus Einstein could still write that the coordinates of
general relativity express the “neighborliness” of events “whose coordinates
differ but little from each other”. As argued in Section 9.1, the assumption
that the physically most meaningful topology of a pseudo-metric space is the
same as the topology of continuous coordinates assigned to that space, even
though there are singularities in the invariant measures based on those
coordinates, is questionable. Given Einstein’s aversion to singularities of
any kind, including even the coordinate singularity at the Schwarzschild
radius, it’s somewhat ironic that he never seems to have worried about the coordinate
singularity of every lightlike interval and the non-transitive nature of
“null separation” in ordinary Minkowski spacetime.

Apparently unconcerned
about the topological implications of Minkowski spacetime, Einstein inferred
from the special theory that “physical reality must be described in terms of
continuous functions in space”. Of course, years earlier he had already
considered some of the possible objections to this point of view. In his 1936
essay on “Physics and Reality” he considered the “already terrifying”
prospect of quantum field theory, i.e., the application of the method of
quantum mechanics to continuous fields with infinitely many degrees of
freedom, and he wrote

To be sure, it has been pointed
out that the introduction of a space-time continuum may be considered as
contrary to nature in view of the molecular structure of everything which
happens on a small scale. It is maintained that perhaps the success of the
Heisenberg method points to a purely algebraical method of description of
nature, that is to the elimination of continuous functions from physics.
Then, however, we must also give up, on principle, the space-time continuum.
It is not unimaginable that human ingenuity will some day find methods which
will make it possible to proceed along such a path. At the present time,
however, such a program looks like an attempt to breathe in empty space.

In his later search for
something beyond general relativity that would encompass quantum phenomena, he
maintained that the theory must be invariant under a group that at least
contains all continuous transformations (represented by the symmetric
tensor), but he hoped to enlarge this group.

It would be most beautiful if
one were to succeed in expanding the group once more in analogy to the step
that led from special relativity to general relativity. More specifically, I
have attempted to draw upon the group of complex transformations of the
coordinates. All such endeavours were unsuccessful. I also gave up an open or
concealed increase in the number of dimensions, an endeavor that … even today
has its adherents.

The reference to complex
transformations is an interesting fore-runner of more recent efforts, notably
Penrose’s twistor program, to exploit the properties of complex functions (cf
Section 9.9). The comment about increasing the number of dimensions certainly
has relevance to current “string theory” research. Of course, as Einstein
observed in an appendix to his Princeton lectures, “In this case one must explain why the
continuum is apparently restricted to four dimensions”. He also
mentioned the possibility of field equations of higher order, but he thought
that such ideas should be pursued “only if there exist empirical reasons to
do so”. On this basis he concluded

We shall limit ourselves to the
four-dimensional space and to the group of continuous real transformations of
the coordinates.

He went on to describe what
he (then) considered to be the “logically most satisfying idea” (involving a
non-symmetric tensor), but added a footnote that revealed his lack of
conviction, saying he thought the theory had a fair probability of being
valid “if the way to an exhaustive description of physical reality on the
basis of the continuum turns out to be at all feasible”. A few years later he
told Abraham Pais that he “was not sure differential geometry was to be the
framework for further progress”, and later still, in 1954, just a year before
his death, he wrote to his old friend Besso (quoted in Section 3.8) that he
considered it quite possible that physics cannot be based on continuous
structures. The dilemma was summed up at the conclusion of his Princeton
lectures, where he said

One can give good reasons why
reality cannot at all be represented by a continuous field. From the quantum
phenomena it appears to follow with certainty that a finite system of finite
energy can be completely described by a finite set of numbers… but this does
not seem to be in accordance with a continuum theory, and must lead to an
attempt to find a purely algebraic theory for the description of reality. But
nobody knows how to obtain the basis of such a theory.

The area of current
research involving “spin networks” might be regarded as attempts to obtain an
algebraic basis for a theory of space and time, but so far these efforts have
not achieved much success. The current field of “string theory” has some
algebraic aspects, but it seems to entail much the same kind of dualism that
Einstein found so objectionable in Lorentz’s theory. Of course, most modern research
into fundamental physics is based on quantum field theory, about which
Einstein was never enthusiastic – to put it mildly. (Bargmann told Pais that
Einstein once “asked him for a private survey of quantum field theory, beginning
with second quantization. Bargman did so for about a month. Thereafter
Einstein’s interest waned.”)

Of all the various
directions that Einstein and others have explored, one of the most intriguing
(at least from the standpoint of relativity theory) was the idea of “expanding
the group once more in analogy to the step that led from special relativity
to general relativity”. However, there are many different ways in which this
might conceivably be done. Einstein referred to allowing complex transformations,
or non-symmetric, or increasing the number of dimensions, etc., but all these
retain the continuum hypothesis. He doesn’t seem to have seriously considered
relaxing this assumption, and allowing completely arbitrary transformations
(unless this is what he had in mind when he referred to an “algebraic
theory”). Ironically in his expositions of general relativity he often proudly
explained that it gave an expression of physical laws valid for completely
arbitrary transformations of the coordinates, but of course he meant
arbitrary only up to diffeomorphism, which in the absolute sense is not very
arbitrary at all.

We mentioned in the
previous section that diffeomorphically equivalent sets can be assigned the
same topology, but from the standpoint of a physical theory it isn't
self-evident which diffeomorphism is the right one (assuming there is one)
for a particular set of physical entities, such as the events of spacetime.
Suppose we're able to establish a 1-to-1 correspondence between certain physical
events and the sets of four real-valued numbers (x0,x1,x2,x3).
(As always, the superscripts are indices, not exponents.) This is already a
very strong supposition, because the real numbers are uncountable, even over
a finite range, so we are supposing that physical events are also
uncountable. However, I've intentionally not characterized these physical
events as points in a certain contiguous region of a smooth continuous
manifold, because the ability to place those events in a one-to-one correspondence
with the coordinate sets does not, by itself, imply any particular arrangement
of those events. (We use the word arrangement here to signify the notions of
order and nearness associated with a specific topology.) In particular, it
doesn't imply an arrangement similar to that of the coordinate sets
interpreted as points in the four-dimensional space denoted by R4.

To illustrate why the
ability to map events with real coordinates does not, by itself, imply a
particular arrangement of those events, consider the coordinates of a single
event, normalized to the range 0-1, and expressed in the form of their
decimal representations, where xmn denotes the nth most
significant digit of the mth coordinate, as shown below

x0
= 0. x01 x02 x03 x04 x05
x06 x07 x08...

x1
= 0. x11 x12 x13 x14 x15
x16 x17 x18 ...

x2
= 0. x21 x22 x23 x24 x25
x26 x27 x28 ...

x3
= 0. x31 x32 x33 x34 x35
x36 x37 x38 ...

We could, as an example,
assign each such set of coordinates to a point in an ordinary four-dimensional
space with the coordinates (y0,y1,y2,y3)
given by the diagonal sets of digits from the corresponding x coordinates,
taken in blocks of four, as shown below

y0
= 0. x01 x12 x23 x34 x05
x16 x27 x38...

y1
= 0. x02 x13 x24 x31 x06
x17 x28 x35 ...

y2
= 0. x03 x14 x21 x32 x07
x18 x25 x35 ...

y3
= 0. x04 x11 x22 x33 x08
x15 x26 x37 ...

We could also transpose
each consecutive pair of blocks, or scramble the digits in any number of
other ways, provided only that we ensure a 1-to-1 mapping. We could even
imagine that the y space has (say) eight dimensions instead of four, and we
could construct those eight coordinates from the odd and even numbered digits
of the four x coordinates. It's easy to imagine numerous 1-to-1 mappings
between a set of abstract events and sets of coordinates such that the actual
arrangement of the events (if indeed they possess one) bears no direct
resemblance to the arrangement of the coordinate sets in their natural space.

So, returning to our task,
we've assigned coordinates to a set of events, and we now wish to assert some
relationship between those events that remains invariant under a particular
kind of transformation of the coordinates. Specifically, we limit ourselves
to coordinate mappings that can be reached from our original x mapping by
means of a simple linear transformation applied on the natural space of x.
In other words, we wish to consider transformations from x to X given by a
set of four continuous functions f i with continuous partial
first derivatives. Thus we have

X0
= f 0 (x0 , x1 , x2 , x3)

X1
= f 1 (x0 , x1 , x2 , x3)

X2
= f 2 (x0 , x1 , x2 , x3)

X3
= f 3 (x0 , x1 , x2 , x3)

Further, we require this
transformation to posses a differentiable inverse, i.e., there exist
differentiable functions Fi such that

x0
= F0 (X0 , X1 , X2 , X3)

x1
= F1 (X0 , X1 , X2 , X3)

x2
= F2 (X0 , X1 , X2 , X3)

x3
= F3 (X0 , X1 , X2 , X3)

A mapping of this kind is
called a diffeomorphism, and two sets are said to be equivalent up to
diffeomorphism if there is such a mapping from one to the other. Any
physical theory, such as general relativity, formulated in terms of tensor
fields in spacetime automatically possess the freedom to choose the
coordinate system from among a complete class of diffeomorphically equivalent
systems. From one point of view this can be seen as a tremendous generality
and freedom from dependence on arbitrary coordinate systems. However, as
noted above, there are infinitely many systems of coordinates that are not
diffeomorphically equivalent, so the limitation to equivalent systems up to
diffeomorphism can also be seen as quite restrictive.

For example, no such
functions can possibly reproduce the digit-scrambling transformations
discussed previously, such as the mapping from x to y, because those mappings
are everywhere discontinuous. Thus we cannot get from x coordinates to y
coordinates (or vice versa) by means of continuous transformations. By
restricting ourselves to differentiable transformations we're implicitly
focusing our attention on one particular equivalence class of coordinate
systems, with no a priori guarantee that this class of systems
includes the most natural parameterization of physical events. In fact, we
don't even know if physical events possess a natural parameterization, or if
they do, whether it is unique.

Recall that the special
theory of relativity assumes the existence and identifiability of a preferred
equivalence class of coordinate systems called the inertial systems. The
laws of physics, according to special relativity, should be the same when
expressed with respect to any inertial system of coordinates, but not
necessarily with respect to non-inertial systems of reference. It was
dissatisfaction with having given a preferred role to a particular class of
coordinate systems that led Einstein to generalize the "gage
freedom" of general relativity, by formulating physical laws in pure
tensor form (general covariance) so that they apply to any system of
coordinates from a much larger equivalence class, namely, those that are
equivalent to an inertial coordinate system up to diffeomorphism. This
entails accelerated coordinate systems (over suitably restricted regions)
that are outside the class of inertial systems. Impressive though this
achievement is, we should not forget that general relativity is still
restricted to a preferred class of coordinate systems, which comprise only an
infinitesimal fraction of all conceivable mappings of physical events,
because it still excludes non-diffeomorphic transformations.

It's interesting to
consider how we arrive at (and agree upon) our preferred equivalence class of
coordinate systems. Even from the standpoint of special relativity the identification
of an inertial coordinate system is far from trivial (even though it's often
taken for granted). When we proceed to the general theory we have a great
deal more freedom, but we're still confined to a single topology, a single
pattern of coherence. How is this coherence apprehended by our senses? Is
it conceivable that a different set of senses might have led us to apprehend
a different coherent structure in the physical world? More to the point,
would it be possible to formulate physical laws in such a way that they
remain applicable under completely arbitrary transformations?